At this point, I introduced a lesson on writing proofs based on an article in NCTM's Mathematics Teacher magazine. The article "Empowering Students' Proof Learning through Communal Engagement" appeared in the volume 109, number 8, April 2016 issue. The lesson followed the suggestions in this article using triangular number formulas.
As suggested in the article, you want to use a formula or concept that students are familiar with. The work with triangular numbers was a perfect choice given the different work students had performed on triangular numbers.
First, we briefly discussed the idea of proof. Proof had previously been discussed in the context of providing a solid mathematical foundation. Students were told that a formula like the triangular number formula appears to work, but how do you know the formula won't break down at some point? Proving the formula always works instills confidence that you can always rely on the formula. It is also not enough to just show a few examples because you don't know if the next example might be the one that doesn't work.
With that, groups were tasked with writing out in words and illustrations why the triangular number formula (recursive or closed) always works. Students were told to think and discuss how they would do this before writing anything on the poster.
This was a difficult task for students. The challenge of actually getting their thinking on paper was much harder than students initially thought. As students worked on their posters I walked around and posed questions or made suggestions to help students clarify their meaning. Specifically, I reminded students that no one would be left behind to explain their poster. The poster had to be self-explanatory.
I initially set a time limit of 15 minutes for the poster task. This was overly optimistic. As time was gradually extended, the total time came closer to 35 minutes. Once posters were completed, I told students to have a notebook to record their thoughts on each of the other posters. Students walked around making notes about which posters proved the formula and what was it about the posters that helped convince them about the truth of the formula. Circling through the posters took approximately 10 minutes.
Students returned to their groups and then discussed their notes. The purpose of the discussions was to identify characteristics of posters that helped prove the formula. I specifically instructed students to focus on the characteristics, not which poster was best. Each group created a list of what they thought should be included in a proof.
At this point we shared out and constructed a class list. Any items added to the list were agreed to by the class. Students were allowed to question or express concerns about any suggested additions. After all the student suggestions were out, I reviewed the list and asked the class permission to add one last item, mathematical terms are used properly. The class agreed this would be okay to add.
I went through this process with two classes. While the individual posters were okay in terms of proving the triangular number formula, students in both classes were able to extract key characteristics. Both classes came out with almost identical ideas. I revised and combined ideas to shorten and clarify the lists. Looking at the result, I was amazed that such a good list was able to be created from the posters I saw.
The result of this lesson was the following list:
- Clear statement of formula or concept being proved.
- Clear explanation why the formula or concept is true and works all the time using specific details and steps in a logical order.
- Clear explanation of what symbols, variables, and labels represent.
- Clear use the formula or concept through examples.
- Supportive illustrations, diagrams, graphs, and tables.
- Correct use of mathematical terms.
Students stayed engaged in this task for approximately 50 minutes. They worked hard on trying to effectively communicate their thinking, on evaluating other posters, and on considering key characteristics that should go into proof.
As the article suggests, this is an effective way to help students build an understanding of what should go into a proof. All of the ideas and thinking came from the students. They weren't told, here's what needs to go into a proof. The students determined from their work and assessment of others' work, what should go into a proof. There is class buy-in as to how future proofs will be evaluated and, the generated list provides a solid foundation on which to further build the concept of proof.
I was pleased about the results and excited about the prospects of building on the work that the classes did today. I am glad I was able to implement the suggestions made in the article and the value that I was able to bring into the classroom as a result of this work.
I did mention to a colleague what I had done in class. She is teaching geometry and is just getting ready to introduce geometric proofs after working through some algebraic proofs. She was interested in what I did and how it went. She is now going to make use of this idea to help her students establish some criteria for proof that can then be transferred to geometric proofs. It's exciting to think of the possibilities for using these ideas in different classes throughout the school.
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